HELP! Vectors - collision

Question

Two boats, A and B, are moving in straight lines and the position of each particle after t seconds is given by the following equations: 𝒓𝐴 = 2𝑡𝒊 + (30 − 𝑡)𝒋 𝒓𝐵 = 2𝑡𝒊 + 60𝒋 − 4𝑡𝒋 (a) Find the initial position vectors for each boat (b) Determine the distance between the boats after 5 seconds (c) Prove that the boats collide, and find the position vector of the point of collision

Answer 1

𝒓𝐴 and 𝒓B represent the position of each boat in the i, j coordinate system as a function of time t.

In a general form these equations look like:

𝒓𝐴(t) = A_x(t) i + A_y(t) j

𝒓B(t) = B_x(t) i + B_y(t) j

The initial position vectors can be found by evaluating each equation at t=0.

You can determine the distance between the boats after 5 seconds by:

-evaluating each equation at t=5 to find the position of each boat 𝒓𝐴(5) and 𝒓B(5)

-finding the distance between the boats along each coordinate axis, Δx = 𝒓𝐴_x(5) - 𝒓B_x(5) and Δy = 𝒓𝐴_y(5) - 𝒓B_y(5)

-plugging the Δx and Δy into the Pythagorean theorem to find the distance between the boats

You can prove the boats collide by:

-setting 𝒓𝐴 and 𝒓B equal to each other and solving for t. Lets call this value t_c for time of collision

-evaluating 𝒓𝐴 and 𝒓B at t_c will give you the position of the boats at t_c

-if 𝒓𝐴(t_c) and 𝒓B(t_c) are equal then you have proved the boats collide, good work

Cheers,

Mike